Conditional quantum entropy

The conditional quantum entropy is an entropy measure used in quantum information theory. It is a generalization of the conditional entropy of classical information theory. For a bipartite state ${\displaystyle \rho ^{AB}}$, the conditional entropy is written ${\displaystyle S(A|B)_{\rho }}$, or ${\displaystyle H(A|B)_{\rho }}$, depending on the notation being used for the von Neumann entropy. The quantum conditional entropy was defined in terms of a conditional density operator ${\displaystyle \rho _{A|B}}$ by Nicolas Cerf and Chris Adami,^[1][2] who showed that quantum conditional entropies can be negative, something that is forbidden in classical physics. The negativity of quantum conditional entropy is a sufficient criterion for quantum non-separability.

In what follows, we use the notation ${\displaystyle S(\cdot )}$ for the von Neumann entropy, which will simply be called "entropy".